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On the uniqueness of solution of magnetostatic vectorpotential problems bythreedimensional finiteelement methodsO. A. Mohammed, W. A. Davis, B. D. Popovic, T. W. Nehl, and N. A. Demerdash Citation: J. Appl. Phys. 53, 8402 (1982); doi: 10.1063/1.330373 View online: http://dx.doi.org/10.1063/1.330373 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v53/i11 Published by the American Institute of Physics. Additional information on J Appl Phys

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  n the uniqueness of solution of magnetostatic vector-potential problems by three dimensional finite elementmethods

O. A. Mohammed W. A. Davis B. D. Popovic, T. W. Nehl, and N. A. Demerdash

Virginia Polytechnic Institute nd State University Blacksburg Virginia 24061

In this paper, particular attention is paid to the impact of finite-element approximation on uniqueness and toapproximations implicit in finite element formulations from the uniqueness requirements standpoint. I t is also

shown that the flux density is unique without qualifications. The theoretical and numerical uniqueness of the

magnetic vector potential in three-dimensional problems is also given. This analysis is restricted to linear,

isotropic media with Dirichlet Boundary conditions. As an interesting consequence of this analysis it is shown

that, under usual conditions adopted in obtaining three-dimensional finite-element solutions, it is not

necessary to specify div A n order that Abe uniquely defined.

PACS numbers: 41.10.Dg

INTRODUCTION

Although for a relatively long time the magnetic

vector-potential has been used for obtaining numericalsolutions of magnetostatic problems, t seems that the

question of uniqueness of the vector-potential i t se l f

has not been treated in deta i l , particularly in con

junction with three··dimensional finite-element solu

t ions. T h i s ~ of c o u r s e ~ is due to the fact that inthe end curl A, and not A i t se l f , i s needed.) In teres tin questions of val idi ty and uniqueness of numerical

3-D f in i te element FE) solutions to magnetostatic

problems has been intensif ied as a resul t of the presentation.and publication of two papers, References

[1] and [2], and their accompanying discussions.This paper addresses the questions of val idi ty and

uniqueness of such 3-D FE solut ions. For simplici ty,we shal l res t r ic t consideration to linear, isotropicmedia with Dirichlet boundary conditions. Both the

curl and divergence nature of the magnetic vector

potential mvp) shall be considered.

STATEMENT OF THE PROBLEM

In th is work, one i s interested in obtaining asolut ion to the magnetostatic form of Maxwell s

equations given by

v X H = J l a )

lb)

vB

=H l c )

where v is the reluctivity of the medium.

I t is well known that B may be expressed as [3]

(2)

where A is the magnetic vector potential mvp). Themvp i s a solut ion to

v X vV X A) = J, (3)

I t is assumed here that A s the exact solut ion which

.satisfies the given boundary conditions. The objectof ~ h e following section i s to consider the uniquenessof A obtained in the solut ion of (3).

THE UNIQUENESS OF THE CURL V X A=BI f Al and A2 are both solutions to (3) with

6A = Al - A2

, then i t follows that

V X v V X cA) = o. 4)

To define the uni9ueness statement, we f i r s t consider

the integral of vlV X aA12. One obtains the following

through integration by parts:

Jv IV X -1 2OA dv =

JloA . V X v (V X oA ]dv

V V

+ 1s loA X v (V X A)] ds (5)

where S i s the given Dirichlet boundary at which

n X A= O. Based on this boundary condition, the surintegral in (5) vanishes. This in tegral would alsovanish for the Neumann condition. Hence, upon substitut ing (4) into (5) one obtains

IvlV X oAI2dv = 0 (6)

V

which, for ~ o s i t i v v, requires that

V X oA = 0 (7)

and by the Helmholtz theorem [4]

oA = v0. (8)

Thus, we see that specifying tangential A on the

boundary surface requires A to be unique to within thegradient of a potential .

The same results may be obtained for the f in i te

element solut ion to 3) . We begin with the followingenergy functional which was previously used in references [1] and [2]:

p U) = J[(U-A) VX vX (D-A)] dv (9)

V

where U is the ~ p p r o x i m a t e solution of (3).

Expanding U as10)

8402 J. Appl. Phys. 53(11), November 1982 0021-8979/82/118402-03 02.40 @ 1982 American Institute of Physics 8402

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with N approximation functions Ui

, we may write the

f i r s t variation of p as follows:

f V i V X (vV X V - A»3c

iV

+ V - A) V X (vV X V. ]1

dv o. 11)

Integrat ing (11) by parts yields

f[2Vi . (V X (vV X V - J)]dv

V f V - A) X (vV X Ui

)

s

(12)

which gives the Euler equation

V X (vV X Ii) = J . (13)

For computational purposes, i t is convenient to

rewrite (12) with Dirichlet conditions as

2 f[VV X Vi . V X V - Vi . J]dv = 0 (14)

V

for i = 1,2, ••• ,N. For l inear approximation functions,

the curls in (14) are p i e c e ~ i s e c o ~ s t a n t s requiring

o n ~ simple in tegrals . I f Al and A2 are two solutionsto U in the numerical problem, then based on (10) and

(14) t is ea'sily shown that

f [vlV X (AI - A2)12

]dV

V

o (15)

and thus A must also be unique to within 10 for the

numerical problem. HoweverL since the curl of 10 is

zero, the solut ion for V X A, that is the flux density

S, is unique with no qualif ications.

THE DIVERGENCE OF THE VECTOR POTENTIAL

In obtaining B as the curl of A as in the pre

vious sect ion, the divergence of A i s not required andthe solut ion for B has been shown above to be unique.

However, in the n u ~ e r i c a l aspect of solving (14) th isnon-uniqueness of A by 10 will cEeate a singularnumerical process for obtaining A which must be solved

by constraining the PEoblem some way. One methodwould be to add (V • U - V • A)2 to the functional

integrand and specify the gauge V • Aas suggested byVan Bladel [5] to sat isfy the sufficiency requirements

of the Helmholtz equation. A second method_would be

to solve the resultant matrix equation for A bypseudoinverse methods [6]. A third method ,.,-hich has

been used successfully in references [1] and [2] i s to

specify n • A (or n • V) n • A» at the boundary sur

face. I t has been found that th is las t technique

gives good resul ts for piecewise l inear , continuous

approximation functions. I t i s the purpose of th issect ion to show that th is uniqueness may be predicted

a prior i and thus a specific gauge need not be im

posed.

A simplist ic view of th is uniqueness may be ob

tained in a numerical form by counting the number of

unknowns and equations (independence assumed). For

the finite-element formulation we obtain three

equations and three unknowns at each in ter ior node.

To complete the problem, we require boundary

conditions on al l three components of the mvp at

boundary nodes. This is consistent with theaddit ional constraint suggested above.

Let us consider Al and A to be s ~ l u t i o n s of (12)

with Dirichlet boundary c o n d i ~ i o n s on A (not jus t

n X A). We wish to show that the difference, '10,between Al and A2 must vanish in V for piecewisel inear, continuous basffi (using tetrahedral elements).

8403 J. Appl. Phys. Vol. 53, No. 11, November 1982

Let us consider an element at the boundary surface.

Since 10 is l inear and must vanish at the boundary,

one can write

(16)

where fil is the boundary surface normal and rthe surface. Taking the curl of 10 we obtainS

is on

(17)

which must be identically zero. Thus CI must be para

l l e l to nl

and we may write (16) as

(18)

I f we describe 10 in an adjacent boundary element with

r constrained to the adjacent edge, thens

(J 9)

I f these elements fa l l along an edge such that

(20)

then cl

and Cz must be zero since 10 must be continuous

at the adjoinlng faces. Hence, these two tetrahedrons

may be deleted from V to determine '10. This resul tmay be generalized to elements with adjacent edges, butnon-adjacent faces. Since for a closed, f in i te volumethere must always be two such tetrahedrons, al l te tra-

hedrons may be deleted to obtain 10 = O. Thus the

V • A n the solut ion is unique (though we have not

determined the value). This resul t may be extended

to the problem with Neumann boundary conditions.

NUMERICAL COMPARISONS

The air-cored coil described in reference [2] has

been solved using the discritization grids given inthat reference. The solution is obtained here with

the boundary conditions A = 0 on the outermost surface

(same as in reference [2]), and also with the condition

on the normal component of A replaced by sett ing the

normal derivat ive to zero (Neumann condition). Table1) shows these results for arbi trar i ly chosen te tra-

hedral elements in the given volume. The results for

the curl and the divergence of the vector potentialare also shown for both of the above cases in thattable. As can be easi ly noticed, the values (s ingleprecis ion on an IBM 370) of the curl of the vector

potential A are the same in ~ o t h cases, hence the curl

is unique. However, there is no fixed pattern for the

divergence of A, and the value of the divergence of Ais inconsequential to the flux densi t ies as expected.

Table 1) Values of Curl ( l ines/sq. inch) andDivergence of the Vector Potentialat Arbitrary Points

fPC' Dirichlet Boundaries I Neumann Boundaries'- '

Ii B : - I - B ; [ " B z - T ' r ~ Y i . - - B ~ ( ' - \ ' l - ' - B ~ P':'X'

1-----"----- . - , - , " j . . I ,i-.- l ~ ~ ~ 4 . L 0.02055. ' 2.7' 64.3 0.0;2058. 6.5

2\ 40 . 4 i - 7. 7 40. 5 57. 5 I 2161. • -0. 7> t .-i j . . I

Y 22.01 -2.1 -2.8 22.5 -2.6i -64 •. -15.1; i \ __ .1.

i 4: 69.31, 1.3; 68.9i -7.5'-190. , -2.2

I ' - - r ~ ~ ~ n 17.8 -13.4 2.51 17.7 -50. 1.0

I 6 220.0\22.5 , -274.( -22.7 ,220.2 ' 22.4;-271. -2.3i

~ ~ ~ . : . ~ U 9 . 71 4 ~ : f -0.9;, 48.91

18.31_-:-44 • -7.0i

~ _ ~ 2 . ~ ~ i 5 4 8 . 5 ~ - 1 6 6 i -4.4; 87.6;547.9:-165., - ~ 2 : 5 :9; - 1 _ ? ' : : l ~ _ ~ 6 _ : 1 L ~ ? - , L _ 8 5 ~ 7 + - : - ~ ~ ~ : , 4 _ ~ _ 2 ~ : ~ + ~ ~ ~ . _ , . . ? ~ ..8

l l ~ , _, 8 9 , ~ L 9 5 . 3 ~ ~ 1 3 . , ; , 2',7; 89.6J 95.1 1-,114. 9.1j

Mohammed et al 8403

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CONCLUSION

A theoretical proof of uniqueness for a three

dimensional f in i te element magnetostatic field solut ionhas been given for a class of 3-D problems. I t has

been shown that the flux density solut ion i s unique

with no qualif ications. The theoretical and numerical

uniqueness of the magnetic vector potential for l inearfinite-elements have also been derived, where the

derivation has been restricted to linear, isotropic

media with Dirichlet boundary conditions. The

consequence of this analysis i s that under usual condit ions assumed in obtaining a three-dimensional

finite-element solution, i t is not_necessary to

explicitly, specify the gauge V • A order to

uniquely define the vector potential A.

REFERENCES

[1] Demerdash N. A., Nehl, T. W. Fouad, F. A. andMohammed O. A., Three Dimensional Finite Element

Vector Potential Formulation of Magnetic Fields inElectrical Apparatus , IEEE Transactions on PowerPower Apparatus and Systems, Vol. PAS-100 No.8,August 1981 pp. 4104-4111.

[2] Demerdash N. A., Nehl, T. W. Mohammed O. A. andFouad, F. A., Experimental Verificat ion and Appli

cation of the Three Dimensional Finite Element Magnetic Vector Potential Method in Electrical Appara

tus , IEEE Transactions on Power Apparatus andSystems, Vol. PAS-100 No.8, August 1981 pp. 4112-

4122.

[3] Strat ton, J. A., Electromagnetic Thoery, McGrawHill, NY 1941, pp. 23ff.

[4] Collin, R. E., Field Theory of Guided WavesMcGraw-Hill, NY 1960, pp. 564-568.

[5] Van Bladel, J . Electromagnetic Fields, McGrawHill, NY 1964, pp. 150-152.

[6] Rao C. R. and S. K. Mitra, Generalized Inverse

of Matrices and I ts Applications, Wiley, NY 1971.

8404 J Appl. Phys. Vol. 53 No. 11 November 1982 Mohammed tal 8404

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